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Periodic motion, displacement, time period, frequency and phase: NEET MCQs with answers

50 questions on this topic of Oscillations and Waves (NEET 2027 syllabus). Here are 12; tap one to see its answer and explanation.

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easy

1. Consider the following two statements regarding periodic motion: Statement I: Every oscillatory motion is necessarily periodic, but every periodic motion need not be oscillatory. Statement II: The motion of the earth around the sun is periodic but not oscillatory. Choose the correct option:

  1. Both Statement I and Statement II are correct.
  2. Statement I is correct but Statement II is incorrect.
  3. Statement I is incorrect but Statement II is correct.
  4. Both Statement I and Statement II are incorrect.
Show answer and explanation

Answer: (A) Both Statement I and Statement II are correct.

Statement I is correct because an oscillatory motion repeats itself about a mean equilibrium position within fixed limits, making it periodic (neglecting damping), while uniform circular motion is periodic without being oscillatory. Statement II is correct because planetary motion repeats at regular intervals of time but does not have a to-and-fro motion about an equilibrium position.

medium

2. Which statement about periodic and oscillatory motion is correct?

  1. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory
  2. Every periodic motion is oscillatory
  3. Uniform circular motion is oscillatory
  4. Periodic motion must be simple harmonic
Show answer and explanation

Answer: (A) Every oscillatory motion is periodic, but every periodic motion need not be oscillatory

Uniform circular motion repeats itself (periodic) but is not a to-and-fro motion about a mean position, so it is not oscillatory.

NCERT Class 11

hard

3. The displacement of a particle executing periodic motion is given by x = 4 cos⁴(ωt/4). The fundamental frequency of this periodic motion is:

  1. ω/π
  2. ω/(2π)
  3. ω/(4π)
  4. 2ω/π
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Answer: (C) ω/(4π)

Using trigonometric identity, cos²(θ) = (1 + cos 2θ)/2, so cos⁴(ωt/4) = [(1 + cos(ωt/2))/2]² = (1/4)[1 + 2cos(ωt/2) + cos²(ωt/2)] = (1/4)[1 + 2cos(ωt/2) + (1 + cos(ωt))/2] = 3/8 + (1/2)cos(ωt/2) + (1/8)cos(ωt). The fundamental angular frequency is the lowest non-zero angular frequency present, which is ω/2. Hence the fundamental frequency is f = (ω/2)/(2π) = ω/(4π).

medium

4. The displacement of an oscillatory system is given by x = A sin³(ωt). This motion can be described as:

  1. Periodic with fundamental frequency ω/2π, but not simple harmonic
  2. Simple harmonic motion with angular frequency 3ω
  3. Simple harmonic motion with angular frequency ω
  4. Non-periodic motion whose amplitude decays with time
Show answer and explanation

Answer: (A) Periodic with fundamental frequency ω/2π, but not simple harmonic

Using the trigonometric identity sin³(θ) = (3 sin θ - sin 3θ)/4, we have x = (3A/4) sin(ωt) - (A/4) sin(3ωt). This is a superposition of two SHMs with angular frequencies ω and 3ω; hence it is a periodic motion with fundamental angular frequency ω (frequency ω/2π), but not a simple harmonic motion itself.

easy

5. The motion of a particle is represented by x(t) = 3 sin(4πt) + 4 sin(4πt + π/2). The amplitude and the frequency of this periodic motion are respectively:

  1. 7 units, 4 Hz
  2. 5 units, 2 Hz
  3. 5 units, 4 Hz
  4. 1 unit, 2 Hz
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Answer: (B) 5 units, 2 Hz

Since 4 sin(4πt + π/2) = 4 cos(4πt), the expression is x(t) = 3 sin(4πt) + 4 cos(4πt). The resultant amplitude is R = √(3² + 4²) = 5 units. The angular frequency is ω = 4π rad/s, so the frequency is f = ω / (2π) = 4π / (2π) = 2 Hz.

hard

6. A particle moves along a straight line such that its displacement from the origin at time t is given by x(t) = A sin(ωt) + B cos(2ωt). Which of the following statements is correct regarding this motion?

  1. The motion is periodic with fundamental period T = 2π/ω, but not simple harmonic.
  2. The motion is simple harmonic with time period T = 2π/ω.
  3. The motion is periodic with fundamental period T = π/ω.
  4. The motion is non-periodic because it involves two distinct frequencies.
Show answer and explanation

Answer: (A) The motion is periodic with fundamental period T = 2π/ω, but not simple harmonic.

The motion is the sum of two harmonic terms with angular frequencies ω and 2ω. Since their frequencies are in the ratio 1:2 (an integer ratio), the combined motion is periodic with the fundamental period corresponding to the lower frequency, T = 2π/ω. However, because it contains more than one frequency component, it is not simple harmonic.

easy

7. A body completes 50 oscillations in 10 s. Its frequency is:

  1. 50 Hz
  2. 500 Hz
  3. 5 Hz
  4. 0.2 Hz
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Answer: (C) 5 Hz

Frequency ν = number of oscillations per second = 50/10 = 5 Hz. The period is T = 1/ν = 0.2 s.

NCERT Class 11

easy

8. The phase difference between the displacement and acceleration of a particle in simple harmonic motion is:

  1. 0 rad
  2. π/2 rad
  3. π rad
  4. 3π/2 rad
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Answer: (C) π rad

In SHM, displacement is x = A sin(ωt + φ) and acceleration is a = -ω²x = ω²A sin(ωt + φ + π). Therefore, acceleration is always opposite in sign to displacement, which corresponds to a phase difference of π radians (180°).

medium

9. A particle executes periodic motion with period T along the x-axis about the origin. Starting from the extreme positive position x = +A at t = 0, the time taken by the particle to travel directly to x = +A/2 is:

  1. T/6
  2. T/12
  3. T/8
  4. T/4
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Answer: (A) T/6

Taking the equation of motion starting from positive extreme as x = A cos(ωt), at x = A/2 we have cos(ωt) = 1/2. Thus ωt = π/3, so (2π/T)t = π/3, giving t = T/6.

medium

10. The displacement of a particle is described by x = 5 sin(4πt + π/3) and that of another particle by y = 5 cos(4πt - π/6), where x and y are in cm and t is in seconds. The phase difference between the motion of the second particle and the first particle is:

  1. π/3 rad
  2. π/2 rad
  3. 0 rad
  4. π rad
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Answer: (C) 0 rad

Rewrite y in terms of sine: y = 5 cos(4πt - π/6) = 5 sin((4πt - π/6) + π/2) = 5 sin(4πt + π/3). Comparing with x = 5 sin(4πt + π/3), both have the same phase angle (4πt + π/3) at all times, so the phase difference is zero.

easy

11. The motion of a particle is given by the function f(t) = cos(ωt) + cos(2ωt) + cos(4ωt). The fundamental time period of this periodic motion is:

  1. 4π/ω
  2. π/ω
  3. π/(2ω)
  4. 2π/ω
Show answer and explanation

Answer: (D) 2π/ω

The angular frequencies present are ω, 2ω, and 4ω. The individual time periods are T₁ = 2π/ω, T₂ = 2π/(2ω) = π/ω, and T₃ = 2π/(4ω) = π/(2ω). The fundamental period of the sum is the lowest common multiple of these periods, which is 2π/ω.

medium

12. Consider the following two statements: Statement I: The motion of the hands of a mechanical clock is periodic but not oscillatory. Statement II: In an oscillatory motion, there is always an equilibrium position about which the body moves back and forth.

  1. Both Statement I and Statement II are correct
  2. Statement I is correct but Statement II is incorrect
  3. Statement I is incorrect but Statement II is correct
  4. Both Statement I and Statement II are incorrect
Show answer and explanation

Answer: (A) Both Statement I and Statement II are correct

The hands of a clock repeat their circular paths at fixed, regular intervals of time, making their motion periodic. However, they do not move to and fro about any stable mean position, so the motion is not oscillatory. Oscillatory motion essentially requires a restoring force or torque directed towards a stable equilibrium position.

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